# Is it possible to collude in a perfect information game?

Is it possible to collude in a perfect information game?

Exercise 10-7 from book "Algorithms And Networking For Computer Games" - Jouni Smed, Harri Hakonen, p 225

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Yes, consider the following situation: Suppose we have two companies $A$ and $B$ that produce oranges. Company $A$ chooses some amount $q_A$ of oranges to produce. Given this information, company $B$ chooses $q_B$ of oranges to produce. The goal of both companies is to maximize their profits. We know that the total quantity is $Q = q_A+ q_B$ and the demand is $f(Q)$. So it is possible for both companies to jointly maximize their profits.

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Yes but you can't collude with your partner in chess. Am I miss something? – user8985 Apr 2 '11 at 22:48
You can collude against other players - in the answer here, the companies are colluding against their customers. In chess, players can collude but typically won't want to because there are no other opponents. Imagine, though, if you are playing a chess match where you both need a draw to progress in a tournament, it's likely you will play for stalemate (to both players' advantage). – Ronald Apr 8 '12 at 11:30
@Ronald: What you in fact do is play into a boring position and agree a draw. This is a common form of collusion in chess tournaments. – Chris Eagle Nov 26 '12 at 12:27

Collusion is one of the major problems with games with more than 2 players: the players that are losing can join forces to bring down the leader. This often becomes more a matter of politics than mathematics, and is therefore difficult to analyse.

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An extensive form is a complete desription of the strategic environment faces. So players have exactly the options available that the game tree presents. In some cases, one might think about some background story in which one can interpret some moves as "colluding", but such an interpretation lies outside game theory.

Edit: It seems the book considers "colluding" as players in a computer game cheating by not following the explicit rules. In a perfect information game, players cannot cheat by exchanging information. In a generic finite game of perfec information, super-rational players can't collude by the usual backward-induction logic. In infite games, they might be able to coordinate by pre-play communication.

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